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**Introduction to & Overview of Statistical & Thermal Physics**

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**Basis of Thermodynamics + Some History**

A Quote from The Feynman Lectures on Physics Volume 1, Chapter 1. Feynman is discussing the fundamental importance of the atomic hypothesis to physics: “If, in some cataclysm, all of scientific knowledge were to be destroyed, & only one sentence passed on to the next generations of creatures…I believe it is the atomic hypothesis…all things are made of atoms; little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another”. Richard P. Feynman ( )

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**Physics I: “Heat is a Form of Energy” James Joule’s Experiment**

Mgh = W = Q & Q = mcT Work can raise the water temperature. James Joule ( ) Joule’s Experimental Setup

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**Carnot: “Engine of Highest Possible Efficiency”**

Nicolas Léonard Sadi Carnot ( ) “Carnot” is the name of a famous French family in politics & science.

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**The Carnot Family President of France, 1887–1894.**

Lazare Nicolas Marguerite Carnot ( ), mathematician & politician. Nicolas Léonard Sadi Carnot ( ), mathematician & eldest son of Lazare, a pioneer of thermodynamics. Hippolyte Carnot ( ), politician & second son of Lazare. Marie François Sadi Carnot ( ), son of Hippolyte, President of France, 1887–1894. Marie Adolphe Carnot ( ), son of Hippolyte, mining engineer & chemist. A number of streets etc. are named after this family throughout France. Nicolas Léonard Sadi Carnot ( )

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**Importance of the Atomic Hypothesis**

Boltzmann's contribution to 19th Century science was vital, but had a tragic outcome! At the end of the 19th century, several puzzling facts (which eventually led to quantum theory), triggered a reaction against 'materialist' science. Some people questioned whether atoms exist. Boltzmann, whose work was based on the atomic concept, found himself cast as their chief defender and the debates became increasingly bitter. Ludwig Boltzmann ( )

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**In 1906, he committed suicide!**

Boltzmann Prone to bouts of depression, Boltzmann came to believe that his life's work had been rejected by the scientific community. This wasn’t true! In 1906, he committed suicide! Boltzmann If despair over rejection, or frustration over being unable to prove his point, were contributing factors the irony would be great indeed. Soon after Boltzmann's death, clinching evidence was found for atoms, & few would ever doubt their existence again.

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**Robert Brown & Brownian Motion**

Observed irregular movement of pollens in water under a microscope. [1st observation of “Brownian motion”: S. Gray, Phil. Trans. 19, 280, (1696).] Brown’s Major Contribution Proved that non-organic particles also have Brownian motion. Thus, Brownian motion is not a manifestation of life. Robert Brown (A botanist!)

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**Einstein, Brownian Motion, & Atomic Hypothesis**

“The Miracle Year” Einstein published 4 papers in the Annalen der Physik in 1905. The Photoelectric Effect Brownian Motion Special Theory of Relativity Which topic was his PhD thesis? Which topic was his Nobel Prize? Albert Einstein 1905

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**Einstein on Thermodynamics**

“A theory is the more impressive the greater the simplicity of its premises, and the more extended its area of applicability”. “Classical thermodynamics… is the only physical theory of universal content which I am convinced that, within the applicability of its basic concepts, will never be overthrown”. Albert Einstein 10

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**Eddington on Thermodynamics**

“If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much the worse for Maxwell’s equations”. “But if your theory is found to be against the second law of thermodynamics I can offer you no hope; there is nothing for it but to collapse in deepest humiliation.” Sir Arthur Eddington, 1929 11 11

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**Introduction to Stat Mech**

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**Basic Definitions & Terminology**

Thermodynamics (“Thermo”) is a macroscopic theory! Thermo ≡ The study of the Macroscopic properties of systems based on a few laws & hypotheses. It results in The Laws of Thermodynamics!

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1. Thermo: Derives relations between the macroscopic properties (& parameters) of a system (heat capacity, temperature, volume, pressure, ..). 2. Thermo: Makes NO direct reference to the microscopic structure of matter. For example, from thermo, we’ll derive later that, for an ideal gas, the heat capacities are related by Cp– Cv = R. But, thermo gives no prescription for calculating numerical values for Cp, Cv. Calculating these requires a microscopic model & statistical mechanics.

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**Kinetic Theory is a microscopic theory!**

. 1. It applies the Laws of Mechanics (Classical or Quantum) to a microscopic model of the individual molecules of a system. 2. It allows the calculation of various Macroscopically measurable quantities on the basis of a Microscopic theory applied to a model of the system. For example, it might be able to calculate the specific heat Cv using Newton’s 2nd Law along with the known force laws between the particles that make up the substance of interest.

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**Kinetic Theory is a microscopic theory!**

. 3. It uses the microscopic equations of motion for individual particles. 4. It uses the methods of Probability & Statistics & the equations of motion of the particles to calculate the (thermal average) Macroscopic properties of a substance.

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**Statistical Mechanics (or Statistical Thermodynamics)**

1. Ignores a detailed consideration of molecules as individuals. 2. Is a Microscopic, statistical approach to the calculation of Macroscopic quantities. 3. Applies the methods of Probability & Statistics to Macroscopic systems with HUGE numbers of particles.

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**Statistical Mechanics**

3. For systems with known energy (Classical or Quantum) it gives BOTH A. Relations between Macroscopic quantities (like Thermo) AND B. NUMERICAL VALUES of them (like Kinetic Theory).

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**This course covers all three! Statistical Mechanics:**

1. Thermodynamics 2. Kinetic Theory 3. Statistical Mechanics Statistical Mechanics: Reproduces ALL of Thermodynamics & ALL of Kinetic Theory. It is more general than either!

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**A Hierarchy of Theories**

(of Systems with a Huge Number of Particles) Statistical Mechanics (the most general theory) ___________|__________ | | Thermodynamics Kinetic Theory (a general, macroscopic theory) (a microscopic theory most easily applicable to gases)

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**Statistical Mechanics:**

Remarks on Statistical & Thermal Physics A brief look at where we are going. A general survey. No worry about details yet! The Key Principle of CLASSICAL Statistical Mechanics: Consider a system containing N particles with 3d positions r1,r2,r3,…rN, & momenta p1,p2,p3,…pN. The system is in Thermal Equilibrium at absolute temperature T. We’ll show that the probability of the system having energy E is: P(E) ≡ e[-E/(kT)]/Z Z ≡ “Partition Function”, T ≡ Absolute Temperature, k ≡ Boltzmann’s Constant

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**The Classsical Partition Function A 6N Dimensional Integral!**

Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT) A 6N Dimensional Integral! This assumes that we have already solved the classical mechanics problem for each particle in the system so that we know the total energy E for the N particles as a function of all positions ri & momenta pi. E = E(r1,r2,r3,…rN,p1,p2,p3,…pN) D Don’t panic! We’ll derive this later!

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**CLASSICAL Statistical Mechanics: P(E) ≡ e[-E/(kT)]/Z**

Let A ≡ any measurable, macroscopic quantity. The thermodynamic average of A ≡ <A>. This is what is measured. Use probability theory to calculate <A> : P(E) ≡ e[-E/(kT)]/Z <A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E) Another 6N Dimensional Integral! Don’t panic! We’ll derive this later!

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**P(En) ≡ exp[-En/(kT)]/Z**

The Key Principle of QUANTUM Statistical Mechanics: Consider a system which can be in any one of N quantum states. The system is in Thermal Equilibrium at absolute temperature T. We’ll show that the probability of the system being in state n with energy En is: P(En) ≡ exp[-En/(kT)]/Z Z ≡ “Partition Function” T ≡ Absolute Temperature k ≡ Boltzmann’s Constant

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**The Quantum Mechanical**

Partition Function Z ≡ ∑nexp[-En/(kT)] Don’t panic! We’ll derive this later!

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**QUANTUM Statistical Mechanics:**

Let A ≡ any measurable, macroscopic quantity. The thermodynamic average of A ≡ <A>. This is what is measured. Use probability theory to calculate <A>. P(En) ≡ exp[(-En/(kT)]/Z <A> ≡ ∑n <n|A|n>P(En) <n|A|n> ≡ Quantum Mechanical expectation value of A in quantum state n. Don’t panic! We’ll derive this later!

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**Question: What is the point of showing this now?**

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**Answer Question: What is the point of showing this now?**

Classical & Quantum Statistical Mechanics both revolve around the calculation of P(E) or P(En).

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**Question: What is the point of showing this now?**

Answer Classical & Quantum Statistical Mechanics both revolve around the calculation of P(E) or P(En). To calculate the probability distribution, we need to calculate the Partition Function Z (similar in the classical & quantum cases).

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**Answer Quoting Richard P. Feynman: “P(E) & Z are at the summit of both**

Question: What is the point of showing this now? Answer Classical & Quantum Statistical Mechanics both revolve around the calculation of P(E) or P(En). To calculate the probability distribution, we need to calculate the Partition Function Z (similar in the classical & quantum cases). Quoting Richard P. Feynman: “P(E) & Z are at the summit of both Classical & Quantum Statistical Mechanics.”

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**The Statistical/Thermal Physics “Mountain”**

Statistical Mechanics (Classical or Quantum) P(E), Z Calculation of Measurable Quantities Equations of Motion The Statistical/Thermal Physics “Mountain”

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**Statistical/Thermal Physics “Mountain”**

P(E), Z Calculation of Measurable Quantities Equations of Motion Statistical/Thermal Physics “Mountain” The entire subject is either the “climb” UP to the summit (calculation of P(E), Z) or the slide DOWN (use of P(E), Z to calculate measurable properties). On the way UP: We’ll rigorously define Thermal Equilibrium & Temperature. On the way DOWN, we’ll derive all of Thermodynamics beginning with microscopic theory.

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